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(1)

New Gauge Symmetries from String Theory

Pei-Ming Ho

Physics Department

National Taiwan University

Sep. 23, 2011@CQSE

(2)

Gauge vs Global

For covariant quantities

Old definitions in some textbooks:

U = independent of spacetime  global U = function of spacetime  gauge (local)

(3)

Gauge vs Global (2)

• Translation symmetry is usually considered as a global symmetry.

• Different interpretation of the transformation:

Translation by a specific length L can be

“gauged” (defined as a gauge symmetry)

 equivalence:

(compactification on a circle).

• Gauge potential is useful but not necessary.

   

x(t) » x(t) + L

(4)

Gauge vs Global (3)

• 2nd example:

base space = R+ with Neumann B.C. at x = 0.

• 3rd example:

space  an interval of length L/2 w.

Neumann B.C.

Space/(subgroups of isometry)  orbifolds

x® - x, f(x) ® f(-x)

(5)

Gauge vs Global (4)

New (better) definition:

• Gauge Symmetry

– Transformation does not change physical states.

– A physical state has multiple descriptions.

– Transformation changes descriptions.

• Global Symmetry

– Transformation changes physical states.

This is a more fundamental distinction than spacetime-dependence.

(6)

Non-Abelian Gauge Symmetry

Modification/Generalization?

   

d A

(1)

= dL

(0)

+[A

(1)

, L

(0)

] F

(2)

= dA

(1)

+ A

(1)

A

(1)

Þ d F

(2)

= [F

(2)

, L

(0)

]

(7)

Non-Commutative Gauge Theory

• D-brane world-volume theory in B-field background. [Chu-Ho 99, Seiberg-Witten 99]

• Commutation relation determined by B-field.

• U(1) gauge theory is non-Abelian.

   

[xi, xj] = iQij

(8)

Lie 3-Algebra Gauge Symmetry

• BLG model for multiple M2-branes [07].

• needed for manifest compatibility with SUSY.

(ABJM model does not have manifest full SUSY.)    

[Ta, Tb, Tc] = f abcdTd A = Aiab(Ta Ä Tb)dxi

(Dif)a = ifa + Aibcfd f bcda

(9)

Generalization of Lie 3-algebra

• All Lie n-algebra gauge symmetries are special cases of ordinary (Lie algebra) non-Abelian

gauge symmetries.

   

[Ta1 , Ta2 ,, Tan] = f a1a2anan+1Tan+1 A = Aa1a2an-1 (Ta1 Ä Ta2 ÄÄ Tan-1 ) f = faTa

[A, f] = Aa1a2an-1fa

n f a1a2anan+1Tan+1

(10)

More Gauge Theories

• Abelian Higher-form gauge theories

• Self-dual gauge theories

• Non-Abelian higher-form gauge theories

• NA SD HF GT w. Lie 3 (on NC space?)

A(n) = n!1 Ai1i2indxi1 Ùdxi2 ÙÙdxin F(n+1) = dA(n)

dA(n) = dL(n-1) dF(n+1) = 0

(11)

Dirac Monopole

• Jacobi identity (associativity)

is violated in the presence of magnetic monopoles.

• Distribution of magnetic monopoles  gauge bundle does not exist 

2-form gauge potential (3-form field strength) (Abelian bundle  Abelian gerbe)

Q: How to generalize to non-Abelian gauge theory?

   

[[Di, Dj ], Dk] +[[Dj, Dk], Di] +[[Dk, Di], Dj] = 0

(12)

Self-Dual Gauge Fields

• Examples:

type IIB supergravity type IIB superstring M5-brane theory twistor theory

• D=4k+2 (4k) for Minkowski (Euclidean) spacetime.

(13)

Self-Dual Gauge Fields

• When D = 2n, the self-duality condition is

• a.k.a. chiral gauge bosons.

• How to produce 1st order diff. eq. from action?

Trick: introduce additional gauge symmetry

   

F = *F

F = n!1 Fi1i2in dxi1 Ù dxi2 ÙÙ dxin

*F = n!1 (D-n)!1 e i1i2iD F i1i2in dxin+1 Ù dxin+2 ÙÙ dxiD

(14)

Chiral Boson in 2D

[Floreanini-Jackiw ’87]

• Self-duality

• Lagrangian

• Gauge symmetry

• Euler-Lagrange eq.

• Gauge transformation

• 1-1 map btw space of sol’s and chiral config’s

   

f  ˙ - ¢  f  = 0

   

L = 12 f ¢ ( ˙ f - ¢ f )

   

x(t -x)f = 0

   

Þ (t -x)f = g(t)

(15)

• Due to gauge symmetry, f  is not an observable.

k=f is an observable  the Lagrangian is

• Formal nonlocality is unavoidable, although physics is local.

• Equivalent to a Weyl spinor in 2D in terms of the density of solitons.

• Lorentz symmetry is hidden.

   

L = 12 k(t, x)

( ò

dye(x- y) ˙ k (t, y) - k(t, x)

)

   

   

[

f

(t, x),

f

(t, y)] = i

e

(x- y)

(16)

Comments

• No vector potential needed for gauge symmetry Vector potential is useful for defining

cov. quantities from deriv. of cov. quantities.

But it is not absolutely necessary.

• The formulation can be generalized to self-dual higher-form gauge theories.   

Dm = m + Am

(17)

Example: M5-brane theory (D=5+1)

[Howe-Sezgin 97, Pasti-Sorokin-Tonin 97, Aganagic-Part- Popescu-Schwarz 97, …]

[Chen-Ho 10]

[Ho-Imamura-Matsuo 08]

(18)

Questions

• What is the geometry of Abelian higher-form gauge symmetry?

(bundles  gerbes?)

• How to define non-Abelian higher-form gauge symmetry?

• Can we generalize the notion of covariant derivative and field strength?

• Do we still need the covariant derivative?

(19)

Non-Abelian Self-Dual 2-Form Gauge Potential

• M5-brane in the (3-form) C-field background.

[Ho-Matsuo 08, Ho-Imamura-Matsuo-Shiba 08]

• Non-Abelian gauge symmetry for 2-form gauge potential

• Nambu-Poisson algebra (ex. of Lie 3-algebra) (Volume-Preserving Diffeomorphism)

• Part of the gauge potential  VPD

• 1 gauge potential for 2 gauge symmetries!

[Ho-Yeh 11]

(20)

Non-Local Non-Abelian

Self-Dual 2-Form Gauge Field

• Need non-locality to circumvent no-go thms for multiple M5-branes. [Ho-Huang-Matsuo 11]

• In “5+1” formulation, decompose all fields

into zero modes vs. non-zero modes in the 5-th dimension.

• Zero modes  1-form potential A in 5D

• Non-zero modes  2-form potential B in 5D

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Comments

• Self-duality: Instantons, Penrose’s twistor theory applied to Maxwell and GR.

• In 4D, 2-form ≈ 0-form, 3-form ≈ (-1)-form, 4-form ≈ trivial through EM duality.

• Expect more new gauge symmetries from string theory to be discovered.

• “Symmetry dictates interaction” -- C. N. Yang.

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